Related papers: Lipschitz stratifications in power-bounded o-minim…

These notes focus on the Lipschitz geometry of sets that are definable in o-minimal structures (expanding the real field). We show that every set which is definable in a polynomially bounded o-minimal structure admits a stratification which…

In a former paper the first and third authors introduced the notion of direction set for a subset of R^n, and showed that the dimension of the common direction set of two subanalytic subsets, called directional dimension, is preserved by a…

We prove that local Lipschitz Killing curvatures of definable sets in a polynomially bounded o-minimal structure are continuous along strata of Whitney stratifications and locally Lipschitz if the stratifications are (w)- regular.

We show that the subgradient method converges only to local minimizers when applied to generic Lipschitz continuous and subdifferentially regular functions that are definable in an o-minimal structure. At a high level, the argument we…

It is well-known that the convergence of a family of smooth functions does not imply the convergence of its gradients. In this work, we show that if the family is definable in an o-minimal structure (for instance semialgebraic, subanalytic,…

We show that functions definable in power bounded $T$-convex fields have the (multidimensional) Jacobian property. Building on work of I. Halupczok, this implies that a certain notion of non-archimedean stratifications is available in such…

A closed subset of $\mathbb{R}^q$, definable in some given o-minimal structure, is Lipschitz normally embedded in $\mathbb{R}^q$ if and only if its one-point compactification is Lipschitz normally embedded in the unit sphere ${\bf S}^q$($ =…

We prove the existence of Verdier stratifications for sets definable in any o-minimal structure on (R, +, .). It is also shown that the Verdier condition (w) implies the Whitney condition (b) in o-minimal structures on (R, +, .). As a…

The paper concerns uniform Yomdin-Gromov parametrizations together with an estimate of their number, which generalizes a theorem by Cluckers-Forey-Loeser to arbitrary equicharacteristic zero valued fields with analytic structure. To this…

We define "t-stratifications", a strong notion of stratifications for Henselian valued fields $K$ of equi-characteristic 0, and prove that they exist. In contrast to classical stratifications in Archimedean fields, t-stratifications also…

Given an o-minimal structure, we show that every definable (in this structure) mapping that is Lipschitz with respect to the inner metric can be approximated by $\mathscr{C}^1$ mappings that are Lipschitz with respect to the inner metric…

We show that certain families of sets in $\mathbb{R}^2$ (or $\mathbb{R}^n$) which are neither definable nor have bounded VC-dimension are nonetheless uniformly approximately definable in the real field, an o-minimal structure.

We investigate Bruhat-Tits buildings and their compactifications by means of Berkovich analytic geometry over complete non-Archimedean fields. For every reductive group G over a suitable non-Archimedean field k we define a map from the…

We consider the problem of minimizing a differentiable function with locally Lipschitz continuous gradient on a stratified set and present a first-order algorithm designed to find a stationary point of that problem. Our assumptions on the…

We first show that the projection image of a discrete definable set is again discrete for an arbitrary definably complete locally o-minimal structure. This fact together with the results in a previous paper implies tame dimension theory and…

Let $K$ be a complete non-Archimedean field $K$ with separated power series, treated in the analytic Denef--Pas language. We prove the existence of definable retractions onto an arbitrary closed definable subset of $K^{n}$, whereby…

We present a general framework, treating Lipschitz domains in Riemannian manifolds, that provides conditions guaranteeing the existence of norming sets and generalized local polynomial reproduction - a powerful tool used in the analysis of…

We prove an analogue of the Yomdin-Gromov Lemma for $p$-adic definable sets and more broadly in a non-archimedean, definable context. This analogue keeps track of piecewise approximation by Taylor polynomials, a nontrivial aspect in the…

Cluckers and Lipshitz have shown that real closed fields equipped with real analytic structure are o-minimal. This generalizes the well-known subanalytic structure $\mathbb{R}_{\mathrm{an}}$ on the real numbers. We extend this line of…

Let G be a connected semisimple group over a non-Archimedean local field. For every faithful, geometrically irreducible linear representation of G we define a compactification of the associated Bruhat-Tits building X(G). This yields a…